Fields
Complex Numbers
Definition: Complex numbers are numbers of the form a+bi where a,b∈R and i is the imaginary unit such that i2=−1. The set of complex numbers is denoted by C.
C={a+bi∣a,b∈R}
addition and multiplication are defined as follows:
(a+bi)+(c+di)=(a+c)+(b+d)i(a+bi)(c+di)=(ac−bd)+(ad+bc)i
The set of real numbers is a subset of the complex numbers, i.e. R⊂C. The set of imaginary numbers is also a subset of the complex numbers, i.e. I⊂C.
Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
eix=cos(x)+isin(x)
where e is Euler's number, the base of natural logarithms, and i is the imaginary unit, which satisfies the equation i2=−1.
The formula is still valid if x is a complex number. In particular, if x=π, Euler's formula states that:
eiπ+1=0
The complex conjugate of a complex number z=a+bi is given by zˉ=a−bi. The complex conjugate of a complex number z is denoted by zˉ.
#EE501 - Linear Systems Theory at METU